Shape and structural relaxation of colloidal tactoids

Facile geometric-structural response of liquid crystalline colloids to external fields enables many technological advances. However, the relaxation mechanisms for liquid crystalline colloids under mobile boundaries remain still unexplored. Here, by combining experiments, numerical simulations and theory, we describe the shape and structural relaxation of colloidal liquid crystalline micro-droplets, called tactoids, where amyloid fibrils and cellulose nanocrystals are used as model systems. We show that tactoids shape relaxation bears a universal single exponential decay signature and derive an analytic expression to predict this out of equilibrium process, which is governed by liquid crystalline anisotropic and isotropic contributions. The tactoids structural relaxation shows fundamentally different paths, with first- and second-order exponential decays, depending on the existence of splay/bend/twist orientation structures in the ground state. Our findings offer a comprehensive understanding on dynamic confinement effects in liquid crystalline colloidal systems and may set unexplored directions in the development of novel responsive materials.

The theory (lines) and the experimental data (symbols) predict the short axis of the tactoids r under various extension rate. The data corresponding to the zero shear where r is equal to its equilibrium value are obtained from the tactoids at equilibrium condition in a cuvette. Note that since the modeling (equation 4) assumes the homogenous internal configuration for the tactoids under the shear rate, at zero extension rate only the data of the tactoids that hold homogenous configuration at equilibrium (or at zero shear extension rate) are presented.

Supplementary Fig. 5 | Schematic showing how the thickness behind every pixel is obtained. a, 2D
schematic of the tactoids as it is seen in retardance images. The green square is a schematic of a pixel that is seen in the retardance images. b, 3D schematic of the tactoids used to calculate the thickness d of the sample behind every pixel. show the aspect ratio (α=R/r) versus volume (V) of the tactoids that are formed in the suspension with a concentration that is set within the isotropic-nematic coexistence region (Table S2). The circle, triangle, and square symbols show homogenous, bipolar, and cholesteric tactoids, respectively. The data at equilibrium are collected from the samples of BLG II (a) and SCNC (b) that are placed in a cuvette. For the equilibrium phase diagram of the BLG I, we refer to our recent study, where we see the transition from homogenous to bipolar at a volume 2,000 µm 3 and bipolar to cholesteric at a volume of 7,000 µm 3 .  Based on an overdamped system with several frictional processes, we consider that the characteristic shape relaxation time, * , is the sum of two contributions anisotropic and isotropic as

Supplementary
where , is characteristic shape relaxation time of elongated isotropic tactoids. In the literature of simple droplets relaxation, , has been well established as (Supplementary Ref. [4][5] where is the interfacial tension, Requiv.= ((r 2 R) 1

)
, with 4 and 5 the Landau constants, and in the terms of Frank elastic constant, it becomes: with K the Frank elastic constant for splay and bending (assumed to be equal) and K2 the Frank twist elastic constant. Furthermore, to reveal the impact of the tactoid's size, the characteristic tactoid size Requiv. is considered in the formulation of .
Before proceeding to carry out the dimensional analysis, we refine the selection of material properties to avoid irrelevant or redundant quantities. For tactoids that relax to nematic phases either homogenous or bipolar, the macroscopic pitch length, ! , does not affect + as for nematic phases and ! = ∞. In such cases, the ! is an irrelevant material property and should be excluded from the dimensional analysis. In addition, Supplementary Ref. 7 has shown that, for cholesteric tactoids, the pitch length depends on the tactoid's size; hence, the ! should also be excluded from the dimension analysis for tactoids that relax to cholesteric phases because of the presence of the characteristic tactoid size in the dimension analysis. In other words, the macroscopic pitch length, ! , should be excluded from the entire dimension analysis.
Applying dimensional analysis on the above-mentioned contributing factors yields where is a positive dimensionless constant and all exponents, i.e. , , , and , can be any real dimensionless number. For simplicity and without loss of generality, we consider Υ = 1 while defining pre-factor b as + = b .
To determine the exponents present in Supplementary Eq. 4, an extensive uncorrelated parametric study for each phase of homogenous nematic (HN), bipolar nematic (BN), and chiral nematic (N*) is performed. Thereafter, the anisotropic contribution, + = * − , , is computed using equations S1-S2 and fitted with the power laws shown in Supplementary Eq. 4. Through regression analysis, the proposed exponents are tabulated in Supplementary Due to the fact that experimental measurements of 8 can be reformulated as The next step is to estimate the mobilities 8 Note that each of Supplementary Eq. 6-8 can turn to be equality equations by use of constant pre-factors, and we can consider all these constants embedded in the main pre-factor b in Supplementary Eq. 9.
Putting all together, characteristic shape relaxation time becomes: that predicts the characteristic shape relaxation time * with a single fitting parameter b.

Supplementary Note 4 Determination of amyloid fibrils and cellulose nanocrystals suspensions properties
Here we provide details on the calculation of the properties of the liquid crystalline systems used in this study, reported in Table 1 in the main text.
where $4 is the Debye length, is the Euler's constant that is 0.577 and is given by (Supplementary with Q the Bjerrum length which is 0.70 nm, k1 the modified Bessel function of the second kind and  Table 1.
Critical concentration. The critical concentrations are measured from the completely phase-separated suspension of the liquid crystalline with a concentration that is set within the isotropic-nematic coexistence region. In Supplementary Table 3, I and N denote the concentration of the isotropic and nematic phases of the phase-separated systems, respectively, that were measured gravimetrically. The volumetric concentration is calculated as where \,hM,N* is the density of the fibrils that is 1. Elastic constants. We used analytical expressions previously proposed to calculate the splay, bend and twist elastic constants (Supplementary Ref. [19][20]. The twist constant 5 is calculated according to Anchoring strength. To obtain anchoring strength ω, we followed Wulff construction estimating ω = (α/2) 2 when anchoring strength is higher than one and = − 1 when anchoring strength is less than or equal to one. Note that here α denotes the aspect ratio (R/r) of the homogenous tactoids in the equilibrium state.

Interfacial tension.
To estimate the interfacial tension of the tactoids we use the universal scaling law where f is a constant equal to 0.3 ( Supplementary References 7 and 22). Note that and L are taken to be equal to Df,m, and Lf,w, respectively.

Viscosity.
We measured the viscosities of the isotropic and nematic phases of the phase-separated liquid crystalline suspensions. We estimate the viscosity of the tactoids to be equal to the viscosity of the nematic phase and the viscosity of the medium phase to equal the viscosity of the isotropic phase. We measured that the viscosities of the liquid crystalline phases are different depending on the shear rate, showing shear thinning behavior. In our calculations in this study, we take the zero shear viscosity values as reported in Table 1

Modeling of the deformation of the tactoids
Here we present modeling of the deformation of the tactoids under external stresses. This has been well documented for simple fluids, but for tactoids, the physics becomes complex due to the energy terms associated with the internal structure of the tactoids and their anisotropic features. We look at the deformation of the droplet under uniaxial flow field with extension rate given εẋx = ∂u A ∂x , where the ux is the flow speed and x is the direction of the motion of the flow (see Supplementary Figure 1). Our approach relies on capturing the energy gained by the tactoids under the external stresses imposed by extensional flow field and incorporating that energy to free-energy landscape of the tactoids that is well described by scaling form of Frank-Oseen elasticity theory.
To start we consider a tactoid under an extensional flow that is at is elongated under a flow field shown in Supplementary Figure 1. To find the normal stresses applied to the tactoid, we first note from εẋx = ∂u A ∂x that the velocity in x-direction is Therefore, from Supplementary Eq. 16-17, one can see that where the value of c0 can be found using kinematic condition stating that the velocity is parallel to the droplet interface at interface, meaning The normal stresses at the interface of the tactoid are obtained as following where popo is the normal stress in r՛ direction and nono is the normal stress in z՛ direction.
where we set the displacement velocity of interface * and Z as * = / and * = / . The terms po and no in capture the area of the tactoids in r՛ and z՛ directions, respectively, as po = ′ and no = ′. Note that here volume of the tactoids kept unchanged during deformation as previously we Additionally, as all three classes of the homogenous, bipolar and cholesteric tactoids hold homogenous configuration under extreme deformation, as can be seen in Figure 1 giving us the steady-state elongated shape of the tactoids under a given extensional flow field.

Supplementary Note 6
Determination of the order parameter The order parameter S is obtained using S= /d∆n0 where is the optical retardance value, d is the thickness of the sample and ∆n0 is the birefringence corresponding to a perfectly aligned nematic phase or when the order parameter is 1 (Supplementary Ref. [26][27]. The retardance value of every pixel within the tactoids is calculated using retardance images taken with LC-PolScope as = (pixel gray value/max range of pixel gray value) × retardance ceiling. The value of d is calculated for every pixel assuming a spinodal shape for the tactoids (see the schematic provided in Supplementary Figure 5 where to capture the structural relaxation of the tactoids. As explained in the main text, we present our calculation independent from ∆n0 as an exact value of ∆n0 is often challenging to obtain experimentally, however is the same for a given liquid crystalline system. MATLAB programing is used to perform the measurements.

Supplementary Note 7
Nematic-cholesteric phase diagram of the tactoids at equilibrium